I was not referring directly to the War Dept., but, in any case, to the extent the War Dept was a hinderance, would probably not have gotten too different an answer between Grant or McClellan.
In the War Dept Halleck was a cypher, Lincolns Chief of Staff. after it was discovered he would, or could, not accept responsibility of command his Generals.
The river crossing that settled little macs fate in Lincolns mind, was how long it took the AoP took to cross the Potomac after Antietam, unopposed, compared to how long it took Lee to get the ANV accross it. That is after McClellan decided to actually cross it.
After Antietam, McClellan was wandering around Norther Va., worried that he might meet the ANV when he was not eady and Grant moved South, no matter what Lee was doing.
Halleck and the War Department may have been more of a problem for McClellan than Grant. McClellan made enemies and acted as if he was in command -- not the President, not Stanton, not Halleck. Since all three of those actually were above him, he had a cantankerous relationship up the chain of command. Grant, OTOH, acknowledged the authority of Lincoln and Stanton (Halleck was not directly above him, but Grant regarded Halleck as a supporter at that point). This is no different than the Confederate side: Lee did a much better job of managing his relationship with Davis and Richmond than Joe Johnston, Bragg and Beauregard did. Grant did a much better job of managing his relationship with Lincoln and Washington than McClellan, Buell and Rosecrans did. Both Lee and Grant got better support as a result.
But we should note that Grant earned the support he got, just as Lee did. The support was not based simply on victories. Grant and Lee worked hand-in-hand with their bosses (really hard to do with a Stanton); they consulted and advised, they argued strongly for their own positions and goals, but in the end they acted as loyal subordinates and earned the trust of the Presidents they served.